- Email: [email protected]
Forced concentration oscillations for catalytic reactions with stop-effect
FORCED CONCENTRATION CATALYTIC REACTIONS
OSCILLATIONS FOR WITH STOP-EFFECT
JAN THLJLLIE’ Polish Academy of Sciences, Institute of Chemical Engineering, 44-100 Gliwice, ul. Baltycka 5, Poland and ALBERT RENKEN Ecole Polytechnique F&d&ale, Institut de gknie chimique, EPFL-Ecublens, Switzerland (Received
18 May 1990; accepted
CH-101s
Lausanne,
13 June 1990)
Abstract-The influence of forced concentration oscillations on a catalytic reaction with stop-effect was studied based on two different adsorption-desorption models. Both models predict mean reaction rates which can be more than twice as high as the maximum rate under optimum steady state conditions. An analytical solution is presented to describe the mean performance as a function of concentration, length of period and cycle split
INTRODUCTION
An interesting phenomenon of rapid increase of reaction rate during deamination of amines on alumina after a stop in the supply of amines was first observed by Hogan (1973). He stopped the diisopropylamine feed into the stream of inert carrier gas in the differential flow reactor propene formation
and found that the reaction rate of was increased 2-3 times compared
to the steady state value. The rise was only temporary and was followed by a decrease to zero, due to the absence of the reactant in the feed. This phenomenon was also observed by Koubek et al. (1980a, b) for the dehydration of alcohols and the deamination of primary amines on acid-base oxide catalysts. The authors called their observations “StopEffect”. The stop-efSect (SE) was explained by two different adsorption-desorption models, suitabIe to describe the experimentally observed transient behaviour of the catalyst after replacing the reactant by an inert gas. Nowobilski and Takoudis (1986) proposed a model assuming two different active sites, whereby the reactant (amine or alcohol) is strongly adsorbed on an acid site (S,) of the catalyst and an empty basic site (S,) is required for reaction to take place. Elementary steps for the model (model I) are:
A+S AS, + $-11-B
2 ZAS, t-2 + C + s, + s,.
(2) (3)
production rate is low in the steady state. After a stop in the flow rate of A, it desorbs from the surface. Supposing that the desorption from the basic sites proceeds much faster than from the acid ones (k_ z 9 k_ I), the concentration of empty sites is rising quickly, which results in an increase of the reaction rate until the accumulated surface compound AS, has been consumed. A different model was proposed by Koubek et al. (1980a, b) to fit their experimental results (model II). According to model II, the reactant can be adsorbed on an occupied site forming a kind of second layer of A molecules and hence stopping the surface reaction. Model II is summarized as follows:
(4)
AS+AgASA k-z AS.2B+C+%
65)
After stopping the reactant feed, the “second layer” bIocking the surface will desorb liberating active AS species. As a result the reaction rate will increase, going through a maximum and finally tending to zero. This interesting observation of performance increase under transient conditions leads to the question whether it could be possible to get mean transformation rates higher than in optimum steady state by repeating the feed-interruption regularly and imposing fluctuations to the system concentration in a square wave fashion as indicated in Fig. 1.
When the adsorption of A is strong on both sites, active sites S, are blocked by A molecules and the ‘Author to whom correspondence should be addressed.
(5)
MATHEMATICAL MODELLING Supposing a differential reactor, the dynamic behaviour of the surface compounds for model I can be 1083
1084
JAN THULLIE and ALBERT RENKEN
described
by the following
equations:
v=kJA][$]-k_,[AS,]-kJAS,][S,I (7) 9
= k2[A][$]
- k_,[AS,]
(8)
where [s,]
= 1 - [AS,];
Quite similar equations d[AS] __ dt
= 1 - [As,].
can be derived
(9)
from model II: Fig. 1. Concentration
= k, [A] [s] - k, [AS] - k3 [As] - k,[A]
w
[s,]
= k,[A]
[AS] + k_,[ASA]
[AS] - k_,[ASA]
(10) (11)
Notation [Xl.
for model
- [ASA].
(12)
dt
for (n - 1)r < t < (n -
1)~ + ye
for (n - 1)~ + it < t < m
B=k,[A] D = E/(E + F)
Z = E/B.
(13) [
l-
The initial condition continuity condition (14) is
is [X] = D for t = 0, and the which follows from eqs (13) and
I+((nm
VI.=&
xexp[: -(E+F)t] +&
l)r+yr) 1,
2, . . , n
(15)
for (n -
= lim[X],+,. f-nr for model
I: for (n - 1)~ < t < (n - 1)~ + y7
= [As,]
(16)
U” - (PT)“_’ U(S - 1) ,(l-_(P-lUT-‘r-‘)_U”S (1 - p-1 UT-‘)
n =
Zlim[YJ, f+“T
exp ( - Ft)
P-‘UT-’
for (n - l)z G 2 < (n - l)t + Yr
= Zlim[Y],
lim [Xl, r-((n-l)r+)q
- 1)
1
x 1 - (P-IuT-‘)“(14)
n = 1, 2, . . .
[Xl.
- DT”-‘(P-‘UT-lS
D(P_‘u)“-’
=
-(E+F)CYI,+B
for (n - 1)r + yr < t < n7
Notation
II:
CA1 (K,,CAl + 1)
E = k,
[Xl” dt
oscillations.
The solutions of eqs (13) and (14) can be found by a similar way as shown by Thullie et al. (1986). The results are:
dEX1,= - F[X],
=
during
F=k_,+k,
Assuming instantaneous equilibrium for the surface species AS, and ASA respectively, eqs (2) and (5), the mean reaction rate under forced concentration oscillations can be investigated. When the reactant concentration changes between zero and [A] the above equations can be reduced to:
dC Yl,
A
for (n - 1)~ G t 6 (n - 1)~ + it
= [AS]
[Y],, = [AS] [s] = 1 - [As]
where
of substance
1 (17)
l)r + y7 <
t < n7
where P = exp (Er); T = exp (Fr);
U = exp (Eyr); S = exp (Fyz).
[Yln=[ASl] F=k_,
for(n-l)t+y~
E = 4 CA1 -
k~&CAllW,,CAI+ 1)
B = k,[A] D = B/(E + F) z=
1.
(18)
Now the mean reaction rate during the nth cycle can be obtained. It takes different forms for each model. MEAN REACTION
The instantaneous defined as:
RATE FROM MODEL
R, = k,[AS,](I
1
rate for this model
reaction -
[As,]).
is
(19)
1085
Catalytic reactions with stop-effect [AS,]
the relaxed steady state:
is given by eqs (16), (17) and
C&l = K,,CAIl(K,,CAl+ l)*HCt -(n - IIT- VI = (20) lirn!? r-0
R,,,
where H[t - (n - l)r - rr] (Heaviside function):
+-
The mean reaction
step
rate over one period
T E”=
a
function
0 for t <(n - 1)r + yr 1 for f 2 (n - 1)~ + yr.
yr] =
H[t - (n - 1)~ -
is
is:
II- irr+yr
1 R,(t)dt r .r (“-1)r
= ;
R,(r) dt
s (n- ljr
r +i
R,(t)dt.
- y)
[ME + (B - E)DFIY(lE(1 - y) + F
one obtains
Q=
for the nth period:
1
x 1 -(Pm’UTm’)“-’
ME(E
+ F) + (B - E)BF
ME(E
+ F) - (B - E)BE’
-s)
(E + F)T
+ 1 - (P-‘UT-‘)“-
1 -P-i
UT-i
1
(p-lUT-,Y-l
1ys-’
P-‘UT-‘)
ti=
n
=
R&(1
Fr
- P-‘uT-‘s)
I_p-1UT-L
For relaxed steady state operation, factor becomes:
* = i&.-0 (*---1)
+ CM- (B - W’I (E + Fk x (1 _ s)(s-l -P- ‘UT_‘) (1 - P-LuT-l)
-
(23)
The mean production rate depends on the cycle split y and the length of period T. For very long periods, the system will attain the steady state condition within each part of the period. The mean production rate reaches a quasi steady state. For z + cc, eq. (23) becomes: R QSS
rate rate’ (2Sa)
When forced cycling is repeated sufFiciently often, the results within the periods become identical and one obtains for n -+ co : “‘cc
time averaged cyclic production maximum steady state production
M = k,D.
where
,imR
Q,(f;” ;) ; F~ . (27)
It can be shown that Q is greater than 1 for E > 0. As the reaction rate under steady state conditions passes a maximum (Thullie and Renken, 1990), the mean rate uni_;r periodic operation will be referred to this maximal value. Hence an enhancement factor is defined as:
- y)
CM-(B--)DlB(l
(35)
in the form
R RSS = Rs (1 - ~1 +
(1 - s-1)
P-‘UT-’
[M - (B - E)D]B(l +--------_ D(E + F) +
(25)
where Rs is the steady state reaction rate obtained for constant concentration [A]. It can easily be proved that the second term in expression (25) is always greater than zero. This result indicates that the mean reaction rate for small cycling times is always greater than in quasi steady state. Introducing
eq. (25) can be rewritten
I-
Y)
(21)
r Jz,n-L)r+yr
After integration
= R,(l
lim R.
(2gb)
The enhancement is shown in Fig. 2 for one set of parameters as a function of concentration of reactant A during the second part of a cycle. It is seen that for an invariant cycle split (v = const), a maximum can occur which is higher than 1. When an optimal cycle split (r = yap) is used, the maximal enhancement attained is up to 12% over the optimum steady state value for [A] = 0.1 (bold line in Fig. 2). The enhancement factor increases when both equilibrium constants K, and K,, decrease and rate constant k, tends to higher values. The optimum split is given by: (E + F)(Q ~ 1) ~ &t;(E Yap =
+ F)(Q - 1)
E(Q - 1) for E > 0
(29a)
yap.= (E + F)(Q- 1) + ,./QF(E _tF2_rQ~_z 1,
=
E(Q- 1)
= k,CA%l,(l - CA%I,)(l - Y)
= R,(l - y).
the enhancement
for E < 0 forEQ>E+F
(24)
For short cycle times a limit of expression (23) can be calculated and the mean reaction rate corresponds to
(29b)
Yap =
1-E
2BZ
k, for E = 0 and [A] > 2k,.
(29c)
JAN THULLIE
1086
and ALBERT RENKEN
Y
model
UP
10
II
,-H
model
0
0.02
0 04
1
0.06
008
/
[Al Fig. 2. Enhancement factor as a function of concentration [A] during the second part of a cycle; model 1. Input data: k, = 1000; k_ t = 0.001; K,, = 100; k, = 0.1.
Figure 3 shows for the same parameter set the optimum split as a function of reactant concentration CA]. ~0, changes quite rapidly for small concentrations ([A] < 0.02). For higher concentrations, the value of yO, is greater than 0.9. This is expected due to the fact that only a short time is required to adsorb enough reactant A on the surface which reacts in the second part of the cycle. When the equilibrium reaction
rate
be calculated
after from
R = k,[AS,],
a stop
= 0.001; K,, = 100; k, = 0.1.
0.12
R 0
assumption is not valid, the in the flow of reactant A can
10
0.04
the relation:
exp
k, + ~CAWsU x (1 - [As,ls
- (k_,
-
+ k3)f
cxp (k-zt))
exp ( - k_,r))
I
0.12
(30)
which is obtained as a solution of an appropriate set of differential equations. Under these conditions the maximum reaction rate is lower compared to that calculated for the equilibrium model. In consequence oscilthe mean reaction rate under concentration lations will be lower than the value predicted by eq. (25). Some examples of the reaction rate behaviour in contrast to the equilibrium case are given in Fig. 4. The time to reach the maximum reaction rate is given as z* = -&~n[2k,&,,,1
(31)
R 0
10
t Fig. 4. Reaction rate after a stop in the flow of reactant A. Input data: k, = 1ooO; k-, = 0.001; K,, = 100; k, = 0.1. (a) Model I. (b) Model II.
where g=k_,
Fig. 3. Optimum split as a function of concentration [A] during the second part of a cycle. Input data: k, = 1000; k_ ,
+k_,+2k,-,/~k_,+k_,)2+4k3k_, (32)
for
[Al, 2 SIG W, - cd.
(33)
If inequality (33) is not satisfied, the increase in the reaction rate after a stop does not take place. For small time t* corresponding to high values of k_,(k_, > 10, Fig. 4a), the equilibrium assumption
can be used and the mean reaction rate is estimated from eq. (23) with cycle times T > t*. Equation (25) gives the upper bound of the enhancement attainable for instantaneous adsorption equilibrium.
Catalytic MEAN REACTION
RATE FOR MODEL
reactions
R, = k, [AS].
Y.p =
(p-lUT-‘)“-’
-
(P-~UT-lS-
+ Y(l
1)
-s-1) 1(1
x 1 - (P-1 UT-‘)“-’ 1 - P-lLJTm’
MB
- y) +
E(E +
F)s
x (s _ II1 -(P-lUT-‘)“-’ 1-
F’l!.JT
- (Pm’UT-I)“S
When
1
(P-1 UT-L).
(35)
n + co lim Rn = E m-CD
Y) -
MB
(1 -
E(E + F)z
P-‘UT~‘S)(l
-s-l)-
1 - Pm’uT-l (34)
For increasing length of period approached and the mean rate steady state kinetics:
y(1
the quasi state is is calculated from
for EC > E + F, and E 2 0. As E is always higher or equal to zero, only one relationship is obtained. Optimal split as a function f[A] is shown in Fig. 3. Compared to model I the optimal split is slightly higher for the same concentrations. The enhancement factor rl, in eq. (28) calculated on the assumption of model II is shown in Fig. 5 and compared to enhancements resulting from model I for optimal cycle split yap. It is interesting to note that the rate at relaxed steady state calculated according to model II (RRSSI,) is always higher than that from model I (R,,,,). Th e relative difference between both models corresponds to about 50% at [A] = 0.1. As discussed for model I, the assumption of instantaneous equilibrium for the second adsorption step results in a maximum enhancement under periodic operation. For non-equilibrium conditions, the reaction rate as a function of time has to be calculated for each period. Due to the fact that the maximum reaction rate after the feed stop is delayed, relaxed steady state is no longer the optimum condition. Nevertheless, mean rates, determined with eq. (36) for high desorption rate constants (k- 2 > 10, Fig. 4b), give a good approximation for the attainable mean reaction rate. The length of period has to be longer than the time needed for the maximum instantaneous rate after a feed stop. This time can be estimated from the rate-time curve calculated according to eq. (41):
R = [AS]exp[ x [exp(
- ii) = k,[Asls(l
At very short cycle times (1 -+ O), eq. (36) is simplified and a relationship for the relaxed steady state results:
r-0
= R,,,
= R,
(1 -
y) + G
- (k_, - k_,t)
+ k, - k-,
+ k,)t]
+ ,_km~‘,“s~‘,s_ 1
- exp [ -(k_,
3
2
+ k3)r]]
(41)
# 0.
- y) = R,(l - y). (37)
lima
+ F)(G - 1) (40)
for k-,
lim R= z-cc
(E + F)(G - 1) - JGF(E E(G - 1)
(34)
Supposing established equilibrium for the secona adsorption step, the time averaged reaction rate can be determined according to eq. (21). After integration one obtains for the nth period:
[
1087
tion [A]:
II
Based on the kinetic model II, the instantaneous rate of reaction within a period is given by:
Qzg
with stop-efkct
E(1 E(1 -y)+F
Y)Y
1 (38)
with The form of eq. (38) is similar to eq. (27) derived for model I. As in the former case, the mean reaction rate for small cycling times is higher compared to quasi steady state values. For the relaxed steady state an optimum cycle split exists which is dependent on the reactant concentra-
1.2 1.0 08 06 04 02 0
0
02
0 04
0 06
0.08
0. I
[Al Fig. 5. Enhancement factor as a function of concentration [A] during the second part of a cycle. Input data: k, = IOOO; k_ 1 = 0.001; k,, = 100; k, = 0.1. Solid lines, model II; dashed line, model I. RRSS(~op~/Ks.,.,.
JAN THULLIEand ALBERTRENKEN
1088
E
2.4 20 F
16
G 1.2
;
0.8
K, KU
0.4
ki M
1,
0
0.02
PI Fig. 6. Comparison of enhancement factors for models I and II. Input data: k, = 10; k-I = 0.001; K,, = 100; k, = 0.1.
; RS R,
PI fm As an example, the transient behaviour is plotted in Fig. 4b. It should be noted that for the same set of parameters, the predicted transient behaviours from both models are hardly distinguishable.
CONCLUSIONS
The two kinetic models studied in this paper are suitable to describe the experimentally observed sudden increase of the reaction rate after a feed stop, called stop-effect (Koubek et al., 1980a, b). The main purpose was to investigate the possibility of getting mean reaction rates under forced concentration oscillations higher than the optimal steady state values. As shown in Fig. 6, the reaction rate under relaxed steady state conditions and applying optimum cycle-split increases monotonically beyond the maximal rate for optimum steady state conditions. The enhancement obtainable depends on the sorption equilibrium constant for the considered adsorption steps. The enhancement factor increases with decreasing adsorption equilibrium constants (Fig. 6) as well as with increasing sorption rates for the second steps, which leads to an adsorption equilibrium. Although both models predict similar steady state and transient reaction rates for identical parameters, there is a remarkable difference for the behaviour under periodic operation. Model II always predicts higher enhancement factors compared to model I. The relative difference can be as high as 50%, as shown in Fig. 6.
NOTATION
CA1
CA%1 B D
dimensionless A concentration dimensionless AS, concentration dimensionless constant, B = k, [A] dimensionless constant, D = B/(E + F) for model I D = E/(E + F) for model II
S T t t*
u Z
dimensionless
constant,
E = kCA1 - k~JG,CAllVGrCAI + 1) for model I E = k, [A] (&,[A] + 1) for model II dimensionless constant, F = k_ I + k, dimensionless constant given by eq. (39) constant given by eq. (32) dimensionless Heaviside step function equilibrium constant, K, = k,/k_, equilibrium constant, K,, = k,/k_, dimensionless rate constant dimensionless constant, M = k,D dimensionless constant, P = exp (Ez) dimensionless constant given by eq. (26) dimensionless steady state reaction rate dimensionless instantaneous reaction rate during nth cycle dimensionless concentration of active sites s dimensionless concentration of active sites si dimensionless constant, S = exp (Fyz) dimensionless constant, T = exp (FT) dimensionless time dimensionless time for the maximum reaction rate after a stop in the supply of reactant A dimensionless constant, U = exp (Eyr) dimensionless constant, Z = 1 for model I Z = E/B for model II
Greek letters cycle split Y cycle time enhancement ;,
factor
Subscripts
max OP LSS RSS
maximum optimum steady state quasi steady state relaxed steady state REFERENCES
Hogan, P., 1973, Ph.D. Thesis, Prague
Inst. Chem. Technol., Prague. Koubek, J., Pasek, J. and Ruzicka, V., 1980a, Exploitation of a nonstationary kinetic phenomenon for the elucidation of surface processes in a catalytic reaction. Proc. 7th International Congress on Catalysis, Tokyo, 78, X53. Koubek, J., Pasek, J. and Ruzicka, V., 1980b, Stationary and nonstationary deactivation of alumina and zeolites in elimination reactions. In Catalyst Deactioation. Elsevier,
Amsterdam. Nowobilski, P. J. and Takoudis,
Ch. G., 1986, Periodic operation of chemical reactor systems: are global improvements attainable? Chem. Engng Commun. 40, 249. Thullie, J. and Renken, A., 1990, Transient hehaviour of catalytic reactions with stop-effect. Chem. Engng Commun. (in press). Thullie, J., Chiao, R. and Rinker, R. G., 1986, Analysis of concentration forcing in heterogeneous catalysis. Chem. Engng Common. 48, 191.
OSCILLATIONS FOR WITH STOP-EFFECT
JAN THLJLLIE’ Polish Academy of Sciences, Institute of Chemical Engineering, 44-100 Gliwice, ul. Baltycka 5, Poland and ALBERT RENKEN Ecole Polytechnique F&d&ale, Institut de gknie chimique, EPFL-Ecublens, Switzerland (Received
18 May 1990; accepted
CH-101s
Lausanne,
13 June 1990)
Abstract-The influence of forced concentration oscillations on a catalytic reaction with stop-effect was studied based on two different adsorption-desorption models. Both models predict mean reaction rates which can be more than twice as high as the maximum rate under optimum steady state conditions. An analytical solution is presented to describe the mean performance as a function of concentration, length of period and cycle split
INTRODUCTION
An interesting phenomenon of rapid increase of reaction rate during deamination of amines on alumina after a stop in the supply of amines was first observed by Hogan (1973). He stopped the diisopropylamine feed into the stream of inert carrier gas in the differential flow reactor propene formation
and found that the reaction rate of was increased 2-3 times compared
to the steady state value. The rise was only temporary and was followed by a decrease to zero, due to the absence of the reactant in the feed. This phenomenon was also observed by Koubek et al. (1980a, b) for the dehydration of alcohols and the deamination of primary amines on acid-base oxide catalysts. The authors called their observations “StopEffect”. The stop-efSect (SE) was explained by two different adsorption-desorption models, suitabIe to describe the experimentally observed transient behaviour of the catalyst after replacing the reactant by an inert gas. Nowobilski and Takoudis (1986) proposed a model assuming two different active sites, whereby the reactant (amine or alcohol) is strongly adsorbed on an acid site (S,) of the catalyst and an empty basic site (S,) is required for reaction to take place. Elementary steps for the model (model I) are:
A+S AS, + $-11-B
2 ZAS, t-2 + C + s, + s,.
(2) (3)
production rate is low in the steady state. After a stop in the flow rate of A, it desorbs from the surface. Supposing that the desorption from the basic sites proceeds much faster than from the acid ones (k_ z 9 k_ I), the concentration of empty sites is rising quickly, which results in an increase of the reaction rate until the accumulated surface compound AS, has been consumed. A different model was proposed by Koubek et al. (1980a, b) to fit their experimental results (model II). According to model II, the reactant can be adsorbed on an occupied site forming a kind of second layer of A molecules and hence stopping the surface reaction. Model II is summarized as follows:
(4)
AS+AgASA k-z AS.2B+C+%
65)
After stopping the reactant feed, the “second layer” bIocking the surface will desorb liberating active AS species. As a result the reaction rate will increase, going through a maximum and finally tending to zero. This interesting observation of performance increase under transient conditions leads to the question whether it could be possible to get mean transformation rates higher than in optimum steady state by repeating the feed-interruption regularly and imposing fluctuations to the system concentration in a square wave fashion as indicated in Fig. 1.
When the adsorption of A is strong on both sites, active sites S, are blocked by A molecules and the ‘Author to whom correspondence should be addressed.
(5)
MATHEMATICAL MODELLING Supposing a differential reactor, the dynamic behaviour of the surface compounds for model I can be 1083
1084
JAN THULLIE and ALBERT RENKEN
described
by the following
equations:
v=kJA][$]-k_,[AS,]-kJAS,][S,I (7) 9
= k2[A][$]
- k_,[AS,]
(8)
where [s,]
= 1 - [AS,];
Quite similar equations d[AS] __ dt
= 1 - [As,].
can be derived
(9)
from model II: Fig. 1. Concentration
= k, [A] [s] - k, [AS] - k3 [As] - k,[A]
w
[s,]
= k,[A]
[AS] + k_,[ASA]
[AS] - k_,[ASA]
(10) (11)
Notation [Xl.
for model
- [ASA].
(12)
dt
for (n - 1)r < t < (n -
1)~ + ye
for (n - 1)~ + it < t < m
B=k,[A] D = E/(E + F)
Z = E/B.
(13) [
l-
The initial condition continuity condition (14) is
is [X] = D for t = 0, and the which follows from eqs (13) and
I+((nm
VI.=&
xexp[: -(E+F)t] +&
l)r+yr) 1,
2, . . , n
(15)
for (n -
= lim[X],+,. f-nr for model
I: for (n - 1)~ < t < (n - 1)~ + y7
= [As,]
(16)
U” - (PT)“_’ U(S - 1) ,(l-_(P-lUT-‘r-‘)_U”S (1 - p-1 UT-‘)
n =
Zlim[YJ, f+“T
exp ( - Ft)
P-‘UT-’
for (n - l)z G 2 < (n - l)t + Yr
= Zlim[Y],
lim [Xl, r-((n-l)r+)q
- 1)
1
x 1 - (P-IuT-‘)“(14)
n = 1, 2, . . .
[Xl.
- DT”-‘(P-‘UT-lS
D(P_‘u)“-’
=
-(E+F)CYI,+B
for (n - 1)r + yr < t < n7
Notation
II:
CA1 (K,,CAl + 1)
E = k,
[Xl” dt
oscillations.
The solutions of eqs (13) and (14) can be found by a similar way as shown by Thullie et al. (1986). The results are:
dEX1,= - F[X],
=
during
F=k_,+k,
Assuming instantaneous equilibrium for the surface species AS, and ASA respectively, eqs (2) and (5), the mean reaction rate under forced concentration oscillations can be investigated. When the reactant concentration changes between zero and [A] the above equations can be reduced to:
dC Yl,
A
for (n - 1)~ G t 6 (n - 1)~ + it
= [AS]
[Y],, = [AS] [s] = 1 - [As]
where
of substance
1 (17)
l)r + y7 <
t < n7
where P = exp (Er); T = exp (Fr);
U = exp (Eyr); S = exp (Fyz).
[Yln=[ASl] F=k_,
for(n-l)t+y~
E = 4 CA1 -
k~&CAllW,,CAI+ 1)
B = k,[A] D = B/(E + F) z=
1.
(18)
Now the mean reaction rate during the nth cycle can be obtained. It takes different forms for each model. MEAN REACTION
The instantaneous defined as:
RATE FROM MODEL
R, = k,[AS,](I
1
rate for this model
reaction -
[As,]).
is
(19)
1085
Catalytic reactions with stop-effect [AS,]
the relaxed steady state:
is given by eqs (16), (17) and
C&l = K,,CAIl(K,,CAl+ l)*HCt -(n - IIT- VI = (20) lirn!? r-0
R,,,
where H[t - (n - l)r - rr] (Heaviside function):
+-
The mean reaction
step
rate over one period
T E”=
a
function
0 for t <(n - 1)r + yr 1 for f 2 (n - 1)~ + yr.
yr] =
H[t - (n - 1)~ -
is
is:
II- irr+yr
1 R,(t)dt r .r (“-1)r
= ;
R,(r) dt
s (n- ljr
r +i
R,(t)dt.
- y)
[ME + (B - E)DFIY(lE(1 - y) + F
one obtains
Q=
for the nth period:
1
x 1 -(Pm’UTm’)“-’
ME(E
+ F) + (B - E)BF
ME(E
+ F) - (B - E)BE’
-s)
(E + F)T
+ 1 - (P-‘UT-‘)“-
1 -P-i
UT-i
1
(p-lUT-,Y-l
1ys-’
P-‘UT-‘)
ti=
n
=
R&(1
Fr
- P-‘uT-‘s)
I_p-1UT-L
For relaxed steady state operation, factor becomes:
* = i&.-0 (*---1)
+ CM- (B - W’I (E + Fk x (1 _ s)(s-l -P- ‘UT_‘) (1 - P-LuT-l)
-
(23)
The mean production rate depends on the cycle split y and the length of period T. For very long periods, the system will attain the steady state condition within each part of the period. The mean production rate reaches a quasi steady state. For z + cc, eq. (23) becomes: R QSS
rate rate’ (2Sa)
When forced cycling is repeated sufFiciently often, the results within the periods become identical and one obtains for n -+ co : “‘cc
time averaged cyclic production maximum steady state production
M = k,D.
where
,imR
Q,(f;” ;) ; F~ . (27)
It can be shown that Q is greater than 1 for E > 0. As the reaction rate under steady state conditions passes a maximum (Thullie and Renken, 1990), the mean rate uni_;r periodic operation will be referred to this maximal value. Hence an enhancement factor is defined as:
- y)
CM-(B--)DlB(l
(35)
in the form
R RSS = Rs (1 - ~1 +
(1 - s-1)
P-‘UT-’
[M - (B - E)D]B(l +--------_ D(E + F) +
(25)
where Rs is the steady state reaction rate obtained for constant concentration [A]. It can easily be proved that the second term in expression (25) is always greater than zero. This result indicates that the mean reaction rate for small cycling times is always greater than in quasi steady state. Introducing
eq. (25) can be rewritten
I-
Y)
(21)
r Jz,n-L)r+yr
After integration
= R,(l
lim R.
(2gb)
The enhancement is shown in Fig. 2 for one set of parameters as a function of concentration of reactant A during the second part of a cycle. It is seen that for an invariant cycle split (v = const), a maximum can occur which is higher than 1. When an optimal cycle split (r = yap) is used, the maximal enhancement attained is up to 12% over the optimum steady state value for [A] = 0.1 (bold line in Fig. 2). The enhancement factor increases when both equilibrium constants K, and K,, decrease and rate constant k, tends to higher values. The optimum split is given by: (E + F)(Q ~ 1) ~ &t;(E Yap =
+ F)(Q - 1)
E(Q - 1) for E > 0
(29a)
yap.= (E + F)(Q- 1) + ,./QF(E _tF2_rQ~_z 1,
=
E(Q- 1)
= k,CA%l,(l - CA%I,)(l - Y)
= R,(l - y).
the enhancement
for E < 0 forEQ>E+F
(24)
For short cycle times a limit of expression (23) can be calculated and the mean reaction rate corresponds to
(29b)
Yap =
1-E
2BZ
k, for E = 0 and [A] > 2k,.
(29c)
JAN THULLIE
1086
and ALBERT RENKEN
Y
model
UP
10
II
,-H
model
0
0.02
0 04
1
0.06
008
/
[Al Fig. 2. Enhancement factor as a function of concentration [A] during the second part of a cycle; model 1. Input data: k, = 1000; k_ t = 0.001; K,, = 100; k, = 0.1.
Figure 3 shows for the same parameter set the optimum split as a function of reactant concentration CA]. ~0, changes quite rapidly for small concentrations ([A] < 0.02). For higher concentrations, the value of yO, is greater than 0.9. This is expected due to the fact that only a short time is required to adsorb enough reactant A on the surface which reacts in the second part of the cycle. When the equilibrium reaction
rate
be calculated
after from
R = k,[AS,],
a stop
= 0.001; K,, = 100; k, = 0.1.
0.12
R 0
assumption is not valid, the in the flow of reactant A can
10
0.04
the relation:
exp
k, + ~CAWsU x (1 - [As,ls
- (k_,
-
+ k3)f
cxp (k-zt))
exp ( - k_,r))
I
0.12
(30)
which is obtained as a solution of an appropriate set of differential equations. Under these conditions the maximum reaction rate is lower compared to that calculated for the equilibrium model. In consequence oscilthe mean reaction rate under concentration lations will be lower than the value predicted by eq. (25). Some examples of the reaction rate behaviour in contrast to the equilibrium case are given in Fig. 4. The time to reach the maximum reaction rate is given as z* = -&~n[2k,&,,,1
(31)
R 0
10
t Fig. 4. Reaction rate after a stop in the flow of reactant A. Input data: k, = 1ooO; k-, = 0.001; K,, = 100; k, = 0.1. (a) Model I. (b) Model II.
where g=k_,
Fig. 3. Optimum split as a function of concentration [A] during the second part of a cycle. Input data: k, = 1000; k_ ,
+k_,+2k,-,/~k_,+k_,)2+4k3k_, (32)
for
[Al, 2 SIG W, - cd.
(33)
If inequality (33) is not satisfied, the increase in the reaction rate after a stop does not take place. For small time t* corresponding to high values of k_,(k_, > 10, Fig. 4a), the equilibrium assumption
can be used and the mean reaction rate is estimated from eq. (23) with cycle times T > t*. Equation (25) gives the upper bound of the enhancement attainable for instantaneous adsorption equilibrium.
Catalytic MEAN REACTION
RATE FOR MODEL
reactions
R, = k, [AS].
Y.p =
(p-lUT-‘)“-’
-
(P-~UT-lS-
+ Y(l
1)
-s-1) 1(1
x 1 - (P-1 UT-‘)“-’ 1 - P-lLJTm’
MB
- y) +
E(E +
F)s
x (s _ II1 -(P-lUT-‘)“-’ 1-
F’l!.JT
- (Pm’UT-I)“S
When
1
(P-1 UT-L).
(35)
n + co lim Rn = E m-CD
Y) -
MB
(1 -
E(E + F)z
P-‘UT~‘S)(l
-s-l)-
1 - Pm’uT-l (34)
For increasing length of period approached and the mean rate steady state kinetics:
y(1
the quasi state is is calculated from
for EC > E + F, and E 2 0. As E is always higher or equal to zero, only one relationship is obtained. Optimal split as a function f[A] is shown in Fig. 3. Compared to model I the optimal split is slightly higher for the same concentrations. The enhancement factor rl, in eq. (28) calculated on the assumption of model II is shown in Fig. 5 and compared to enhancements resulting from model I for optimal cycle split yap. It is interesting to note that the rate at relaxed steady state calculated according to model II (RRSSI,) is always higher than that from model I (R,,,,). Th e relative difference between both models corresponds to about 50% at [A] = 0.1. As discussed for model I, the assumption of instantaneous equilibrium for the second adsorption step results in a maximum enhancement under periodic operation. For non-equilibrium conditions, the reaction rate as a function of time has to be calculated for each period. Due to the fact that the maximum reaction rate after the feed stop is delayed, relaxed steady state is no longer the optimum condition. Nevertheless, mean rates, determined with eq. (36) for high desorption rate constants (k- 2 > 10, Fig. 4b), give a good approximation for the attainable mean reaction rate. The length of period has to be longer than the time needed for the maximum instantaneous rate after a feed stop. This time can be estimated from the rate-time curve calculated according to eq. (41):
R = [AS]exp[ x [exp(
- ii) = k,[Asls(l
At very short cycle times (1 -+ O), eq. (36) is simplified and a relationship for the relaxed steady state results:
r-0
= R,,,
= R,
(1 -
y) + G
- (k_, - k_,t)
+ k, - k-,
+ k,)t]
+ ,_km~‘,“s~‘,s_ 1
- exp [ -(k_,
3
2
+ k3)r]]
(41)
# 0.
- y) = R,(l - y). (37)
lima
+ F)(G - 1) (40)
for k-,
lim R= z-cc
(E + F)(G - 1) - JGF(E E(G - 1)
(34)
Supposing established equilibrium for the secona adsorption step, the time averaged reaction rate can be determined according to eq. (21). After integration one obtains for the nth period:
[
1087
tion [A]:
II
Based on the kinetic model II, the instantaneous rate of reaction within a period is given by:
Qzg
with stop-efkct
E(1 E(1 -y)+F
Y)Y
1 (38)
with The form of eq. (38) is similar to eq. (27) derived for model I. As in the former case, the mean reaction rate for small cycling times is higher compared to quasi steady state values. For the relaxed steady state an optimum cycle split exists which is dependent on the reactant concentra-
1.2 1.0 08 06 04 02 0
0
02
0 04
0 06
0.08
0. I
[Al Fig. 5. Enhancement factor as a function of concentration [A] during the second part of a cycle. Input data: k, = IOOO; k_ 1 = 0.001; k,, = 100; k, = 0.1. Solid lines, model II; dashed line, model I. RRSS(~op~/Ks.,.,.
JAN THULLIEand ALBERTRENKEN
1088
E
2.4 20 F
16
G 1.2
;
0.8
K, KU
0.4
ki M
1,
0
0.02
PI Fig. 6. Comparison of enhancement factors for models I and II. Input data: k, = 10; k-I = 0.001; K,, = 100; k, = 0.1.
; RS R,
PI fm As an example, the transient behaviour is plotted in Fig. 4b. It should be noted that for the same set of parameters, the predicted transient behaviours from both models are hardly distinguishable.
CONCLUSIONS
The two kinetic models studied in this paper are suitable to describe the experimentally observed sudden increase of the reaction rate after a feed stop, called stop-effect (Koubek et al., 1980a, b). The main purpose was to investigate the possibility of getting mean reaction rates under forced concentration oscillations higher than the optimal steady state values. As shown in Fig. 6, the reaction rate under relaxed steady state conditions and applying optimum cycle-split increases monotonically beyond the maximal rate for optimum steady state conditions. The enhancement obtainable depends on the sorption equilibrium constant for the considered adsorption steps. The enhancement factor increases with decreasing adsorption equilibrium constants (Fig. 6) as well as with increasing sorption rates for the second steps, which leads to an adsorption equilibrium. Although both models predict similar steady state and transient reaction rates for identical parameters, there is a remarkable difference for the behaviour under periodic operation. Model II always predicts higher enhancement factors compared to model I. The relative difference can be as high as 50%, as shown in Fig. 6.
NOTATION
CA1
CA%1 B D
dimensionless A concentration dimensionless AS, concentration dimensionless constant, B = k, [A] dimensionless constant, D = B/(E + F) for model I D = E/(E + F) for model II
S T t t*
u Z
dimensionless
constant,
E = kCA1 - k~JG,CAllVGrCAI + 1) for model I E = k, [A] (&,[A] + 1) for model II dimensionless constant, F = k_ I + k, dimensionless constant given by eq. (39) constant given by eq. (32) dimensionless Heaviside step function equilibrium constant, K, = k,/k_, equilibrium constant, K,, = k,/k_, dimensionless rate constant dimensionless constant, M = k,D dimensionless constant, P = exp (Ez) dimensionless constant given by eq. (26) dimensionless steady state reaction rate dimensionless instantaneous reaction rate during nth cycle dimensionless concentration of active sites s dimensionless concentration of active sites si dimensionless constant, S = exp (Fyz) dimensionless constant, T = exp (FT) dimensionless time dimensionless time for the maximum reaction rate after a stop in the supply of reactant A dimensionless constant, U = exp (Eyr) dimensionless constant, Z = 1 for model I Z = E/B for model II
Greek letters cycle split Y cycle time enhancement ;,
factor
Subscripts
max OP LSS RSS
maximum optimum steady state quasi steady state relaxed steady state REFERENCES
Hogan, P., 1973, Ph.D. Thesis, Prague
Inst. Chem. Technol., Prague. Koubek, J., Pasek, J. and Ruzicka, V., 1980a, Exploitation of a nonstationary kinetic phenomenon for the elucidation of surface processes in a catalytic reaction. Proc. 7th International Congress on Catalysis, Tokyo, 78, X53. Koubek, J., Pasek, J. and Ruzicka, V., 1980b, Stationary and nonstationary deactivation of alumina and zeolites in elimination reactions. In Catalyst Deactioation. Elsevier,
Amsterdam. Nowobilski, P. J. and Takoudis,
Ch. G., 1986, Periodic operation of chemical reactor systems: are global improvements attainable? Chem. Engng Commun. 40, 249. Thullie, J. and Renken, A., 1990, Transient hehaviour of catalytic reactions with stop-effect. Chem. Engng Commun. (in press). Thullie, J., Chiao, R. and Rinker, R. G., 1986, Analysis of concentration forcing in heterogeneous catalysis. Chem. Engng Common. 48, 191.